In 1842, Samuel Earnshaw showed (in the article “On the nature of the molecular forces which regulate the constitution of the luminiferous ether” published by the journal Transactions of the Cambridge Philosophical Society, Vol. 7, pages 97-112), that magnetic “instability cannot be removed by [any] arrangement” of magnets. In his published proof, Earnshaw examined the potential energy of a single particle being attracted by many others. He showed that the energy field for this particle must always be the shape of a peak or at best a hyperboloid (a saddle), so that motion of a particle in this field will always be unstable. Therefore any arrangement of magnets to attempt to create a focal point in space will be unstable. This instability is of importance to physicians attempting to deliver drugs or other therapies with magnetizable nanoparticles as carriers focused to tumors or to other tissues of interest.
Although originally proved for particles that were permanently magnetized, Earnshaw's result also applies to temporarily magnetizable (e.g., ferromagnetic, ferrimagnetic, paramagnetic, super-paramagnetic) particles immersed in a magnetic field. This proof is recited in the article by A. Nacev et al published in the journal IEEE Control Systems Magazine, Vol. 32, issue 3, and entitled “Towards Control of Magnetic Fluids in Patients: Directing Therapeutic Nanoparticles to Disease Locations.” Specifically, equation S2 of the article shows that the second derivative of the potential energy (i.e., the Laplacian) is negative if the magnetic force on a particle in a magnetic field is in the same direction as the increasing gradient of that field. This negative second derivative implies that at best the particles may reside in an energy saddle, with no stable well that is able to focus or confine the particles. As a result, it is not conventionally possible to concentrate magnetizable particles in an interior volume solely through the use of magnetic fields. The mathematical proof, discussed in the article by A. Nacev et al, illustrates a potential loop-hole in Earnshaw's theorem: if the particle was diamagnetic (i.e., if the magnetic force was in the opposite direction from an increasing magnetic gradient), an energy well could be constructed of externally-applied magnetic fields that could be used to focus the diamagnetic particles at a distance away from the magnetic field source.
The force on a particle immersed in a magnetic field is approximately proportional to the particle's susceptibility. Unfortunately, most diamagnetic materials have very small magnetic susceptibility (and hence experience lower magnetic forces), requiring extremely strong magnetic fields for particle manipulation. M. D. Simon and A. K. Geim, in their 2000 article in the Journal of Applied Physics (Vol. 87, number 9, pages 6200-6204) entitled “Diamagnetic levitation: Flying frogs and floating magnets”, point out that a 12-Tesla magnet is required for droplets of water (or frogs containing water) to overcome gravity. Water has a susceptibility of about 10−5 in CGS units of cm3 mol−1. Ferromagnetic and paramagnetic materials have much higher susceptibilities. Nickel oxides, for example, have susceptibilities on the order of 10,000 cm3 mol−1(i.e., nine orders of magnitude larger).